There seem to be an uncountable number of different equations for calculating the air-to-fuel ratio of an engine's combustion process from its exhaust emissions despite the fact that all of the equations are based on the same chemistry and physics of the same combustion phenomena. Authors such as D'Alleva, Spindt, Brettschneider, Lange, Simons, Stivender, Holl and others have published papers that are often referenced as sources for these equations.
D'Alleva wrote the earliest paper regularly cited in the literature. He described the relationship between the exhaust gas composition and the air/fuel ratio. He published charts that could be used to read the A/F ratio based on exhaust concentrations, according the fuel h/c ratio. This was in 1936, before computers and calculators, so such charts were common and necessary in engineering practice.
Eltinge improved on D'Alleva's charts to include incomplete combustion. The charts could also be used without on O.sub.2 measurement, but then no estimate of measurement error from the size of a triangle formed on the chart by the intersection of the three measurement lines was available.
Spindt published the next major step forward. He published an actual formula using CO, CO.sub.2, HC and O.sub.2. It did not require an assumption of complete combustion. Spindt worked for a fuel company, Gulf Oil, so he was sensitive to the fact that combustion was not complete and that the exhaust gas contained a mix of hydrocarbons related in a complicated way to the fuel and the operating mode of the engine.
In 1973, William Holl at AC Spark plug published formulae that did not require a measurement for oxygen. Since the formulae are algebraically complex and he was interested in making real-time calculations at a time when laboratory computers were not so powerful or easy to program, he developed simplified equations by using power series approximations and ignoring the higher order terms. The simpler forms in use today are variations of this idea.
Brettschneider was next in 1979. He added terms to Spindt's equation to account for both water in the ambient air and to incorporate a measured NO.sub.x into the equation, so it no longer needed to be assumed to be 0. He also included terms for oxygenated fuels. He worked for Bosch, so he was sensitive to the importance of A/F for the performance of carburetors and fuel injection systems. His equation is an evolutionary improvement on Spindt and should replace it.
Other investigators such as Piken and Rouf had in the meantime taken Spindt's ideas and developed equations that did not need the O.sub.2 measurement. When O.sub.2 was present, they proposed using it as validity check. They later extended their result, as did Brettschneider, to include NO.sub.x and H.sub.2 O.
Next came Simons from the TUEV in 1974. He recognized that the extra degree of freedom provided by an O.sub.2 measurement could be used to calculate the equilibrium constant K instead of as a validity check on the other measurements. This improved the agreement of his formula to measured test data. It showed that K could vary widely, and that it was generally lower than the 3.5 that was commonly assumed.
Recently, Mitsubishi investigators Fukui, Tamura, Omori, Saitoh, apparently unaware of the work of Brettschneider and Simons, improved on the Spindt formula by including NO and water vapor. More significantly, they also noted that the equilibrium constant seemed to be modified by the action of the catalyst. They recommended only using the engine gasses for determining A/F. The Simons equation may have performed better with post catalyst measurements, but they did not investigate it.
The equations noted above can be somewhat complex. There are a number of assumptions that are made of the values of physical constants, some chooices about how to use the information that is available, and there is a good deal of flexibility in the algebraic forms that are used to represent the equation.
The physical constants are generally well known, but not precisely known. A slightly different value, when taken into a formula and used in algebraic reformations to calculate other constants, results in formulas with different coefficients. Everyone can recognize that 20.95 and 20.9 are just two slightly different values for the assumed concentration of oxygen in air. But when this number is used to get mole proportions between nitrogen and oxygen, it is not so easy to recognize that 4.77418 and 3.7733 are both. derived from this same physical constant. It is also apparent that the constants are used with varying numbers of significant digits.
Most of the differences between equations are a matter of algebra. Since many expressions arise during the derivation of the A/F equation, there is much room for creativity in the selection of the algebraic steps taken in the simplification process and in the final form of the simplified result.
The very same equation can be expressed using algebraic forms that are so different they can no longer be recognized as equivalent.
Another source of differences are the basic assumptions about what will be significant to include in the calculation. For example:
It is important to include the humidity of the ambient air? PA1 Is there significant water in the fuel itself? PA1 Can the contribution of NO.sub.x be ignored? PA1 Can it be assumed that all the NO.sub.x is NO? PA1 What is the number of carbon atoms on a molecule of HC in the exhaust? Is it the same as in the fuel? PA1 What is the water/gas equilibrium constant?3.8, 3.5, 3.2, or much smaller? PA1 Can it be assumed that the cooler dries the sample completely?
Theoretically, if combustion is complete (HC=0), if NO.sub.x can be neglected, and if the water/gas equilibrium is assumed, a measurement of just CO2 and CO are enough to determine the air/fuel ratio.
If combustion is not complete, one can add a measurement of HC and an assumption about how many molecules of C are in an HC molecule in the exhaust gas. The hydrocarbon molecules are generally smaller in the exhaust than they are in the fuel (except for CNG). Today, combustion is often nearly complete, so under commonly accepted assumptions and for fair accuracy, it is only necessary to measure CO, CO2 and HC.
To be more accurate, a measurement of NO.sub.x and an assumption about how much was NO and how much NO.sub.2 can be applied. Usually, it is assumed NO.sub.2 =0. This may be undesirable for diesel vehicles, but the effect of NO.sub.x on the result is small in any case.
With these measurements of HC, CO, CO.sub.2 and NO.sub.x, the formulae vary only as to whether they include terms for water in the ambient air and fuel, and what values they assume for fundamental constants.
Another dimension opens up however, when a measurement of O.sub.2 is available. This measurement provides another degree of freedom. The degree of freedom can be used in a number of ways. It can eliminate the need for one of the basic assumptions, or it can be used as an error check on the accuracy and consistency of the other measurements. The Simons' method uses it to eliminate the need to assume a value for the water-gas equilibrium constant K. This is perhaps the most practical use.
In determining A/F, equations are usually set up for a fixed measuring configuration, where the values. to be measured and the assumptions to be made are known well ahead of time and are not expected to change. However, a given measurement system often must adapt itself to different situations, depending on customer preference and available measurements. Therefore, rather than a single equation calculation type, it is desirable to have a measurement system that includes an algorithm which calculates the air/fuel ratio under these different circumstances.